If the product of two numbers is 315 and their lcm is 105 what is their HCF

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GCF of 45 and 105 is the largest possible number that divides 45 and 105 exactly without any remainder. The factors of 45 and 105 are 1, 3, 5, 9, 15, 45 and 1, 3, 5, 7, 15, 21, 35, 105 respectively. There are 3 commonly used methods to find the GCF of 45 and 105 - prime factorization, Euclidean algorithm, and long division.

What is GCF of 45 and 105?

Answer: GCF of 45 and 105 is 15.

Explanation:

The GCF of two non-zero integers, x(45) and y(105), is the greatest positive integer m(15) that divides both x(45) and y(105) without any remainder.

Methods to Find GCF of 45 and 105

The methods to find the GCF of 45 and 105 are explained below.

• Using Euclid's Algorithm
• Prime Factorization Method
• Long Division Method

GCF of 45 and 105 by Euclidean Algorithm

As per the Euclidean Algorithm, GCF(X, Y) = GCF(Y, X mod Y)
where X > Y and mod is the modulo operator.

Here X = 105 and Y = 45

• GCF(105, 45) = GCF(45, 105 mod 45) = GCF(45, 15)
• GCF(45, 15) = GCF(15, 45 mod 15) = GCF(15, 0)
• GCF(15, 0) = 15 (∵ GCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of GCF of 45 and 105 is 15.

GCF of 45 and 105 by Prime Factorization

Prime factorization of 45 and 105 is (3 × 3 × 5) and (3 × 5 × 7) respectively. As visible, 45 and 105 have common prime factors. Hence, the GCF of 45 and 105 is 3 × 5 = 15.

GCF of 45 and 105 by Long Division

GCF of 45 and 105 is the divisor that we get when the remainder becomes 0 after doing long division repeatedly.

• Step 1: Divide 105 (larger number) by 45 (smaller number).
• Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (45) by the remainder (15).
• Step 3: Repeat this process until the remainder = 0.

The corresponding divisor (15) is the GCF of 45 and 105.

☛ Also Check:

GCF of 45 and 105 Examples

1. Example 1: For two numbers, GCF = 15 and LCM = 315. If one number is 105, find the other number.

   Solution:

   Given: GCF (x, 105) = 15 and LCM (x, 105) = 315 ∵ GCF × LCM = 105 × (x) ⇒ x = (GCF × LCM)/105 ⇒ x = (15 × 315)/105 ⇒ x = 45

   Therefore, the other number is 45.

Show Solution >

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The GCF of 45 and 105 is 15. To calculate the GCF (Greatest Common Factor) of 45 and 105, we need to factor each number (factors of 45 = 1, 3, 5, 9, 15, 45; factors of 105 = 1, 3, 5, 7, 15, 21, 35, 105) and choose the greatest factor that exactly divides both 45 and 105, i.e., 15.

How to Find the GCF of 45 and 105 by Long Division Method?

To find the GCF of 45, 105 using long division method, 105 is divided by 45. The corresponding divisor (15) when remainder equals 0 is taken as GCF.

If the GCF of 105 and 45 is 15, Find its LCM.

GCF(105, 45) × LCM(105, 45) = 105 × 45 Since the GCF of 105 and 45 = 15 ⇒ 15 × LCM(105, 45) = 4725 Therefore, LCM = 315

☛ GCF Calculator

What are the Methods to Find GCF of 45 and 105?

There are three commonly used methods to find the GCF of 45 and 105.

• By Euclidean Algorithm
• By Long Division
• By Prime Factorization

How to Find the GCF of 45 and 105 by Prime Factorization?

To find the GCF of 45 and 105, we will find the prime factorization of the given numbers, i.e. 45 = 3 × 3 × 5; 105 = 3 × 5 × 7. ⇒ Since 3, 5 are common terms in the prime factorization of 45 and 105. Hence, GCF(45, 105) = 3 × 5 = 15

☛ Prime Number

What is the Relation Between LCM and GCF of 45, 105?

The following equation can be used to express the relation between Least Common Multiple and GCF of 45 and 105, i.e. GCF × LCM = 45 × 105.

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Given:

LCM of numbers = 315

HCF of numbers = 5

Sum of the numbers = 80

Concept:

(LCM × HCF) of two numbers = Product of the two numbers.

Calculation:

 Let the 1st number be ‘a’

2nd number = ‘b’.

∵ a + b = 80

⇒ b = 80 – a

∵ LCM × HCF = Product of two numbers

⇒ 315 × 5 = a × (80 – a)

⇒ 1575 = 80a – a2

⇒ a2 – 80a + 1575 = 0

⇒ a2 – 45a – 35a + 1575 = 0

⇒ a(a – 45) – 35(a – 45) = 0

⇒ (a – 35)(a – 45) = 0

⇒ a = 35 or a = 45

∴ Difference of the numbers = 45 – 35 = 10

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